Could both of these be good ideas?
Common sense says No.
Uncommon sense says Yes, if in vitro experiments, which show that stomach acid plus free fatty acids are powerful germ killers, model what happens in the lumen of the stomach full of food (foods contain substances which could help or hinder the formation of the structures responsible for killing bacteria).
Yes – milk needs fat so that the free fatty acids plus stomach acid can kill harmful organisms potentially in the milk and other foods in the meal.
Yogurt contains beneficial organisms – so we believe – that we do not want to kill. Thus, non-fat yogurt, either by itself or as part of a very low fat meal, would be more beneficial than full-fat yogurt, since stomach acid without free fatty acids is a weak germ killer, about a thousand times weaker.
Probiotic supplements with enteric coating is another way to bypass both stomach acid and fat when the supplements are taken on an empty stomach or with a very low fat meal. (fatty acids plus bile acids are a pretty good germ killer too – bile acids by themselves not so good a germ killer).